Suppose a random variable \(Y\) has the following probability distribution: \(\operatorname{Pr}(Y=1)=p, \operatorname{Pr}(Y=2)=q\), and \(\operatorname{Pr}(Y=3)=1-p-q\). A random sample of size \(n\) is drawn from this distribution, and the random variables are denoted \(Y_{1}, Y_{2}, \ldots, Y_{n}\).

a. Derive the likelihood function for the parameters \(p\) and \(q\).

b. Derive formulas for the MLE of \(p\) and \(q\).